As for optimal locally repairable codes with small distances, the length of optimal locally repairable codes could be much larger than the alphabet size \(q\) .
Arbitrary long optimal locally repairable codes can be constructed by cyclic codes for distance \(d=3, 4\)  [1]}, and the length of an optimal \(q\) -ary locally repairable code is at most roughly \(O(dq^3)\)  for distance \(d\ge 5\)  [2]}. Jin provided an explicit construction of \(q\) -ary optimal locally repairable codes with length \(O(q^2)\)  via binary constant-weight codes for distance 5 and 6 [3]}. From extreme graph theory, there exists explicit construction of optimal locally repairable codes with super-linear length for distance \(d\ge 7\)  [4]}.
Over the past few years, locally repairable codes have been generalized to correct multiple erasure errors [5]}, [6]}, [7]}, [8]} and correct erasures from multiple recovery sets [9]}, [10]}, [11]}.
